Page - 81 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

7.2 Improved Algorithm An especially challenging aspect of the integration procedure presented in chapter 5 is theretentionof fullnumericalprecision in thedeterminationof the integratedpositions and velocities of eqs. (5.2.3) and (5.2.4). For longer arc lengths the numeric values of especially the integrated positionsrinte can become very large. As these integrals are accumulated, the numerical resolution of a standard double precision floating point number can cease to be sufficient to hold all of the necessary information, leading to a loss of precision in the least significant digits. One promising approach to regaining this precision is to split the integral into two parts: The first part is numerically large, and should be solved analytically. The second part of the integral should be smaller in magnitude, and is integrated numerically. This can be seen as a more general formulation of the well-known Encke method for perturbed orbit propagation (Encke, 1852), which was also independently developed by Bond (Bond, 1849). Writing the larger part of the integral to be due to a reference forcef0(τ), and the smaller due to a perturbing force∆f(τ), the acceleration due to their sum is the full acceleration r¨(τ)=f(τ)=f0(τ)+∆f(τ) . (7.2.1) Equivalently to the original equation of motion, this partitioned equation can also be integrated. The reference acceleration is, along with its integrals, equivalent to eqs. (5.1.6) to (5.1.8): r¨ref(τ)=f0(τ) (7.2.2) r˙ref(τ)= r˙ref,0+T ∫ τ 0 f0(τ ′)dτ′ (7.2.3) rref(τ)=rref,0+ r˙ref,0(τT)+T2 ∫ τ 0 (τ−τ′)f0(τ′)dτ′ . (7.2.4) In these equations, the initial values of the reference motion yref,0= [ rref,0 r˙ref,0 ] (7.2.5) appear. The reference forcef0 should be chosen in such a way that the integrals in eqs. (7.2.2) to (7.2.4) have analytical solutions. Computing the complete integrals of both the reference force and accelerations due to perturbing forces gives the true motion of the spacecraft, defined by r¨(τ)=f0(τ)+∆f(τ) , (7.2.6) r(τ)= r˙0+T ∫ τ 0 [ f0(τ ′)+∆f(τ′) ] dτ′ , (7.2.7) r(τ)=r0+ r˙0(τT)+T2 ∫ τ 0 (τ−τ′)[f0(τ′)+∆f(τ′)]dτ′ . (7.2.8) 7.2 Improved Algorithm 81